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Sequence-to-Sequence (seq2seq) tasks transcribe the input sequence to a target sequence. The Connectionist Temporal Classification (CTC) criterion is widely used in multiple seq2seq tasks. Besides predicting the target sequence, a side product of CTC is to predict the alignment, which is the most probable input-long sequence that specifies a hard aligning relationship between the input and target units. As there are multiple potential aligning sequences (called paths) that are equally considered in CTC formulation, the choice of which path will be most probable and become the predicted alignment is always uncertain. In addition, it is usually observed that the alignment predicted by vanilla CTC will drift compared with its reference and rarely provides practical functionalities. Thus, the motivation of this work is to make the CTC alignment prediction controllable and thus equip CTC with extra functionalities. The Bayes risk CTC (BRCTC) criterion is then proposed in this work, in which a customizable Bayes risk function is adopted to enforce the desired characteristics of the predicted alignment. With the risk function, the BRCTC is a general framework to adopt some customizable preference over the paths in order to concentrate the posterior into a particular subset of the paths. In applications, we explore one particular preference which yields models with the down-sampling ability and reduced inference costs. By using BRCTC with another preference for early emissions, we obtain an improved performance-latency trade-off for online models. Experimentally, the proposed BRCTC reduces the inference cost of offline models by up to 47% without performance degradation and cuts down the overall latency of online systems to an unseen level.